mathlib docs: data.nat.gcd

data.nat.gcd

nat.coprime.coprime_div_left

nat.coprime.coprime_div_right

nat.coprime.coprime_dvd_left

nat.coprime.coprime_dvd_right

nat.coprime.coprime_mul_left

nat.coprime.coprime_mul_left_right

nat.coprime.coprime_mul_right

nat.coprime.coprime_mul_right_right

nat.coprime.dvd_of_dvd_mul_left

nat.coprime.dvd_of_dvd_mul_right

nat.coprime.eq_one_of_dvd

nat.coprime.gcd_both

nat.coprime.gcd_eq_one

nat.coprime.gcd_left

nat.coprime.gcd_mul

nat.coprime.gcd_mul_left_cancel

nat.coprime.gcd_mul_left_cancel_right

nat.coprime.gcd_mul_right_cancel

nat.coprime.gcd_mul_right_cancel_right

nat.coprime.gcd_right

nat.coprime.mul

nat.coprime.mul_dvd_of_dvd_of_dvd

nat.coprime.mul_right

nat.coprime.pow

nat.coprime.pow_left

nat.coprime.pow_right

nat.coprime.symm

nat.coprime_div_gcd_div_gcd

nat.coprime_mul_iff_left

nat.coprime_mul_iff_right

nat.coprime_of_dvd

nat.coprime_of_dvd'

nat.coprime_one_left

nat.coprime_one_left_iff

nat.coprime_one_right

nat.coprime_one_right_iff

nat.coprime_self

nat.coprime_zero_left

nat.coprime_zero_right

nat.dvd_gcd_iff

nat.dvd_lcm_left

nat.dvd_lcm_right

nat.eq_zero_of_gcd_eq_zero_left

nat.eq_zero_of_gcd_eq_zero_right

nat.exists_coprime

nat.exists_coprime'

nat.gcd_dvd_gcd_mul_left

nat.gcd_dvd_gcd_mul_left_right

nat.gcd_dvd_gcd_mul_right

nat.gcd_dvd_gcd_mul_right_right

nat.gcd_dvd_gcd_of_dvd_left

nat.gcd_dvd_gcd_of_dvd_right

nat.gcd_dvd_left

nat.gcd_dvd_right

nat.gcd_eq_left

nat.gcd_eq_left_iff_dvd

nat.gcd_eq_right

nat.gcd_eq_right_iff_dvd

nat.gcd_gcd_self_left_left

nat.gcd_gcd_self_left_right

nat.gcd_gcd_self_right_left

nat.gcd_gcd_self_right_right

nat.gcd_le_left

nat.gcd_le_right

nat.gcd_mul_dvd_mul_gcd

nat.gcd_mul_lcm

nat.gcd_mul_left

nat.gcd_mul_left_left

nat.gcd_mul_left_right

nat.gcd_mul_right

nat.gcd_mul_right_left

nat.gcd_mul_right_right

nat.gcd_one_right

nat.gcd_pos_of_pos_left

nat.gcd_pos_of_pos_right

nat.lcm_one_left

nat.lcm_one_right

nat.lcm_zero_left

nat.lcm_zero_right

nat.not_coprime_of_dvd_of_dvd

nat.pow_dvd_pow_iff

nat.prod_dvd_and_dvd_of_dvd_prod

theorem nat.gcd_dvd (m n : ℕ) :

m.gcd n ∣ m ∧ m.gcd n ∣ n

theorem nat.gcd_dvd_left (m n : ℕ) :

m.gcd n ∣ m

theorem nat.gcd_dvd_right (m n : ℕ) :

m.gcd n ∣ n

theorem nat.gcd_le_left {m : ℕ} (n : ℕ) :

0 < m → m.gcd n ≤ m

theorem nat.gcd_le_right (m : ℕ) {n : ℕ} :

0 < n → m.gcd n ≤ n

theorem nat.dvd_gcd {m n k : ℕ} :

k ∣ m → k ∣ n → k ∣ m.gcd n

theorem nat.dvd_gcd_iff {m n k : ℕ} :

k ∣ m.gcd n ↔ k ∣ m ∧ k ∣ n

theorem nat.gcd_comm (m n : ℕ) :

m.gcd n = n.gcd m

theorem nat.gcd_eq_left_iff_dvd {m n : ℕ} :

m ∣ n ↔ m.gcd n = m

theorem nat.gcd_eq_right_iff_dvd {m n : ℕ} :

m ∣ n ↔ n.gcd m = m

theorem nat.gcd_assoc (m n k : ℕ) :

(m.gcd n).gcd k = m.gcd (n.gcd k)

@[simp]

theorem nat.gcd_one_right (n : ℕ) :

n.gcd 1 = 1

theorem nat.gcd_mul_left (m n k : ℕ) :

(m * n).gcd (m * k) = m * n.gcd k

theorem nat.gcd_mul_right (m n k : ℕ) :

(m * n).gcd (k * n) = m.gcd k * n

theorem nat.gcd_pos_of_pos_left {m : ℕ} (n : ℕ) :

0 < m → 0 < m.gcd n

theorem nat.gcd_pos_of_pos_right (m : ℕ) {n : ℕ} :

0 < n → 0 < m.gcd n

theorem nat.eq_zero_of_gcd_eq_zero_left {m n : ℕ} :

m.gcd n = 0 → m = 0

theorem nat.eq_zero_of_gcd_eq_zero_right {m n : ℕ} :

m.gcd n = 0 → n = 0

theorem nat.gcd_div {m n k : ℕ} :

k ∣ m → k ∣ n → (m / k).gcd (n / k) = m.gcd n / k

theorem nat.gcd_dvd_gcd_of_dvd_left {m k : ℕ} (n : ℕ) :

m ∣ k → m.gcd n ∣ k.gcd n

theorem nat.gcd_dvd_gcd_of_dvd_right {m k : ℕ} (n : ℕ) :

m ∣ k → n.gcd m ∣ n.gcd k

theorem nat.gcd_dvd_gcd_mul_left (m n k : ℕ) :

m.gcd n ∣ (k * m).gcd n

theorem nat.gcd_dvd_gcd_mul_right (m n k : ℕ) :

m.gcd n ∣ (m * k).gcd n

theorem nat.gcd_dvd_gcd_mul_left_right (m n k : ℕ) :

m.gcd n ∣ m.gcd (k * n)

theorem nat.gcd_dvd_gcd_mul_right_right (m n k : ℕ) :

m.gcd n ∣ m.gcd (n * k)

theorem nat.gcd_eq_left {m n : ℕ} :

m ∣ n → m.gcd n = m

theorem nat.gcd_eq_right {m n : ℕ} :

n ∣ m → m.gcd n = n

@[simp]

theorem nat.gcd_mul_left_left (m n : ℕ) :

(m * n).gcd n = n

@[simp]

theorem nat.gcd_mul_left_right (m n : ℕ) :

n.gcd (m * n) = n

@[simp]

theorem nat.gcd_mul_right_left (m n : ℕ) :

(n * m).gcd n = n

@[simp]

theorem nat.gcd_mul_right_right (m n : ℕ) :

n.gcd (n * m) = n

@[simp]

theorem nat.gcd_gcd_self_right_left (m n : ℕ) :

m.gcd (m.gcd n) = m.gcd n

@[simp]

theorem nat.gcd_gcd_self_right_right (m n : ℕ) :

m.gcd (n.gcd m) = n.gcd m

@[simp]

theorem nat.gcd_gcd_self_left_right (m n : ℕ) :

(n.gcd m).gcd m = n.gcd m

@[simp]

theorem nat.gcd_gcd_self_left_left (m n : ℕ) :

(m.gcd n).gcd m = m.gcd n

theorem nat.lcm_comm (m n : ℕ) :

m.lcm n = n.lcm m

theorem nat.lcm_zero_left (m : ℕ) :

0.lcm m = 0

theorem nat.lcm_zero_right (m : ℕ) :

m.lcm 0 = 0

theorem nat.lcm_one_left (m : ℕ) :

1.lcm m = m

theorem nat.lcm_one_right (m : ℕ) :

m.lcm 1 = m

theorem nat.lcm_self (m : ℕ) :

m.lcm m = m

theorem nat.dvd_lcm_left (m n : ℕ) :

m ∣ m.lcm n

theorem nat.dvd_lcm_right (m n : ℕ) :

n ∣ m.lcm n

theorem nat.gcd_mul_lcm (m n : ℕ) :

m.gcd n * m.lcm n = m * n

theorem nat.lcm_dvd {m n k : ℕ} :

m ∣ k → n ∣ k → m.lcm n ∣ k

theorem nat.lcm_assoc (m n k : ℕ) :

(m.lcm n).lcm k = m.lcm (n.lcm k)

@[instance]

def nat.decidable (m n : ℕ) :

decidable (m.coprime n)

Equations

m.decidable n = _.mpr (nat.decidable_eq (m.gcd n) 1)

theorem nat.coprime.gcd_eq_one {m n : ℕ} :

m.coprime n → m.gcd n = 1

theorem nat.coprime.symm {m n : ℕ} :

n.coprime m → m.coprime n

theorem nat.coprime_of_dvd {m n : ℕ} :

(∀ (k : ℕ), 1 < k → k ∣ m → ¬k ∣ n) → m.coprime n

theorem nat.coprime_of_dvd' {m n : ℕ} :

(∀ (k : ℕ), k ∣ m → k ∣ n → k ∣ 1) → m.coprime n

theorem nat.coprime.dvd_of_dvd_mul_right {m n k : ℕ} :

k.coprime n → k ∣ m * n → k ∣ m

theorem nat.coprime.dvd_of_dvd_mul_left {m n k : ℕ} :

k.coprime m → k ∣ m * n → k ∣ n

theorem nat.coprime.gcd_mul_left_cancel {k : ℕ} (m : ℕ) {n : ℕ} :

k.coprime n → (k * m).gcd n = m.gcd n

theorem nat.coprime.gcd_mul_right_cancel (m : ℕ) {k n : ℕ} :

k.coprime n → (m * k).gcd n = m.gcd n

theorem nat.coprime.gcd_mul_left_cancel_right {k m : ℕ} (n : ℕ) :

k.coprime m → m.gcd (k * n) = m.gcd n

theorem nat.coprime.gcd_mul_right_cancel_right {k m : ℕ} (n : ℕ) :

k.coprime m → m.gcd (n * k) = m.gcd n

theorem nat.coprime_div_gcd_div_gcd {m n : ℕ} :

0 < m.gcd n → (m / m.gcd n).coprime (n / m.gcd n)

theorem nat.not_coprime_of_dvd_of_dvd {m n d : ℕ} :

1 < d → d ∣ m → d ∣ n → ¬m.coprime n

theorem nat.exists_coprime {m n : ℕ} :

0 < m.gcd n → (∃ (m' n' : ℕ), m'.coprime n' ∧ m = m' * m.gcd n ∧ n = n' * m.gcd n)

theorem nat.exists_coprime' {m n : ℕ} :

0 < m.gcd n → (∃ (g m' n' : ℕ), 0 < g ∧ m'.coprime n' ∧ m = m' * g ∧ n = n' * g)

theorem nat.coprime.mul {m n k : ℕ} :

m.coprime k → n.coprime k → (m * n).coprime k

theorem nat.coprime.mul_right {k m n : ℕ} :

k.coprime m → k.coprime n → k.coprime (m * n)

theorem nat.coprime.coprime_dvd_left {m k n : ℕ} :

m ∣ k → k.coprime n → m.coprime n

theorem nat.coprime.coprime_dvd_right {m k n : ℕ} :

n ∣ m → k.coprime m → k.coprime n

theorem nat.coprime.coprime_mul_left {k m n : ℕ} :

(k * m).coprime n → m.coprime n

theorem nat.coprime.coprime_mul_right {k m n : ℕ} :

(m * k).coprime n → m.coprime n

theorem nat.coprime.coprime_mul_left_right {k m n : ℕ} :

m.coprime (k * n) → m.coprime n

theorem nat.coprime.coprime_mul_right_right {k m n : ℕ} :

m.coprime (n * k) → m.coprime n

theorem nat.coprime.coprime_div_left {m n a : ℕ} :

m.coprime n → a ∣ m → (m / a).coprime n

theorem nat.coprime.coprime_div_right {m n a : ℕ} :

m.coprime n → a ∣ n → m.coprime (n / a)

theorem nat.coprime_mul_iff_left {k m n : ℕ} :

(m * n).coprime k ↔ m.coprime k ∧ n.coprime k

theorem nat.coprime_mul_iff_right {k m n : ℕ} :

k.coprime (m * n) ↔ k.coprime m ∧ k.coprime n

theorem nat.coprime.gcd_left (k : ℕ) {m n : ℕ} :

m.coprime n → (k.gcd m).coprime n

theorem nat.coprime.gcd_right (k : ℕ) {m n : ℕ} :

m.coprime n → m.coprime (k.gcd n)

theorem nat.coprime.gcd_both (k l : ℕ) {m n : ℕ} :

m.coprime n → (k.gcd m).coprime (l.gcd n)

theorem nat.coprime.mul_dvd_of_dvd_of_dvd {a n m : ℕ} :

m.coprime n → m ∣ a → n ∣ a → m * n ∣ a

theorem nat.coprime_one_left (n : ℕ) :

theorem nat.coprime_one_right (n : ℕ) :

theorem nat.coprime.pow_left {m k : ℕ} (n : ℕ) :

m.coprime k → (m ^ n).coprime k

theorem nat.coprime.pow_right {m k : ℕ} (n : ℕ) :

k.coprime m → k.coprime (m ^ n)

theorem nat.coprime.pow {k l : ℕ} (m n : ℕ) :

k.coprime l → (k ^ m).coprime (l ^ n)

theorem nat.coprime.eq_one_of_dvd {k m : ℕ} :

k.coprime m → k ∣ m → k = 1

@[simp]

theorem nat.coprime_zero_left (n : ℕ) :

0.coprime n ↔ n = 1

@[simp]

theorem nat.coprime_zero_right (n : ℕ) :

n.coprime 0 ↔ n = 1

@[simp]

theorem nat.coprime_one_left_iff (n : ℕ) :

1.coprime n ↔ true

@[simp]

theorem nat.coprime_one_right_iff (n : ℕ) :

n.coprime 1 ↔ true

@[simp]

theorem nat.coprime_self (n : ℕ) :

n.coprime n ↔ n = 1

def nat.prod_dvd_and_dvd_of_dvd_prod {m n k : ℕ} :

k ∣ m * n → {d // k = ↑(d.fst) * ↑(d.snd)}

Represent a divisor of m * n as a product of a divisor of m and a divisor of n.

Equations

nat.prod_dvd_and_dvd_of_dvd_prod H = (λ (_x : ℕ) (h0 : k.gcd m = _x), _x.cases_on (λ (h0 : k.gcd m = 0), eq.rec (λ (H : 0 ∣ m * n) (h0 : 0.gcd m = 0), eq.rec (λ (H : 0 ∣ 0 * n) (h0 : 0.gcd 0 = 0), ⟨(⟨0, nat.prod_dvd_and_dvd_of_dvd_prod._proof_1⟩, ⟨n, _⟩), _⟩) _ H h0) _ H h0) (λ (n_1 : ℕ) (h0 : k.gcd m = n_1.succ), (λ (h0 : k.gcd m = n_1.succ), ⟨(⟨k.gcd m, _⟩, ⟨k / k.gcd m, _⟩), _⟩) h0) h0) (k.gcd m) nat.prod_dvd_and_dvd_of_dvd_prod._proof_7

theorem nat.gcd_mul_dvd_mul_gcd (k m n : ℕ) :

k.gcd (m * n) ∣ k.gcd m * k.gcd n

theorem nat.coprime.gcd_mul (k : ℕ) {m n : ℕ} :

m.coprime n → k.gcd (m * n) = k.gcd m * k.gcd n

theorem nat.pow_dvd_pow_iff {a b n : ℕ} :

0 < n → (a ^ n ∣ b ^ n ↔ a ∣ b)

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chain

defs

erase_dup

forall2

func

indexes

intervals

min_max

nat_antidiagonal

nodup

of_fn

pairwise

palindrome

perm

range

rotate

sections

sigma

sort

tfae

zip

matrix

basic

notation

pequiv

multiset

antidiagonal

basic

erase_dup

finset_ops

fold

functor

intervals

lattice

nat_antidiagonal

nodup

pi

powerset

range

sections

sort

nat

basic

cast

choose

digits

dist

enat

fib

gcd

modeq

multiplicity

pairing

parity

prime

sqrt

totient

num

basic

bitwise

lemmas

prime

option

basic

defs

padics

hensel

padic_integers

padic_norm

padic_numbers

pfunctor

multivariate

M

W

basic

univariate

M

basic

pnat

basic

factors

intervals

xgcd

polynomial

degree

basic

algebra_map

basic

coeff

degree

derivative

div

eval

field_division

identities

induction

integral_normalization

monic

monomial

ring_division

qpf

multivariate

constructions

cofix

comp

const

fix

prj

quot

sigma

basic

univariate

basic

rat

basic

cast

denumerable

floor

meta_defs

order

sqrt

real

basic

cardinality

cau_seq

cau_seq_completion

conjugate_exponents

ennreal

ereal

golden_ratio

hyperreal

irrational

nnreal

pi

seq

computation

parallel

seq

wseq

set

intervals

basic

disjoint

image_preimage

ord_connected

unordered_interval

basic

countable

disjointed

enumerate

finite

function

lattice

setoid

basic

partition

sigma

basic

stream

basic

string

basic

defs

zmod

basic

zsqrtd

basic

gaussian_int

W

bool

char

dfinsupp

erased

fin

fin2

fin_enum

finmap

hash_map

holor

indicator_function

lazy_list2

mllist

monoid_algebra

mv_polynomial

opposite

pequiv

pfun

prod

quot

rel

semiquot

subtype

sum

support

sym

sym2

tree

typevec

ulift

vector2

deprecated

group

ring

subgroup

submonoid

dynamics

circle

rotation_number

translation_number

fixed_points

basic

topology

periodic_pts

field_theory

algebraic_closure

chevalley_warning

finite

fixed

minimal_polynomial

mv_polynomial

normal

perfect_closure

separable

splitting_field

subfield

tower

geometry

algebra

lie_group

euclidean

basic

circumcenter

monge_point

triangle

manifold

basic_smooth_bundle

charted_space

local_invariant_properties

mfderiv

real_instances

smooth_manifold_with_corners

times_cont_mdiff

group_theory

perm

cycles

sign

submonoid

basic

membership

operations

abelianization

congruence

coset

eckmann_hilton

free_abelian_group

free_group

group_action

monoid_localization

order_of_element

presented_group

quotient_group

semidirect_product

subgroup

sylow

linear_algebra

affine_space

basic

combination

finite_dimensional

independent

char_poly

coeff

direct_sum

finsupp

tensor_product

adic_completion

basic

basis

bilinear_form

char_poly

contraction

determinant

dimension

direct_sum_module

dual

finite_dimensional

finsupp

finsupp_vector_space

invariant_basis_number

lagrange

linear_action

linear_pmap

matrix

multilinear

nonsingular_inverse

projection

quadratic_form

sesquilinear_form

smodeq

special_linear_group

tensor_algebra

tensor_product

logic

function

basic

conjugate

iterate

basic

embedding

nontrivial

relation

relator

unique

measure_theory

category

Meas

ae_eq_fun

bochner_integration

borel_space

content

decomposition

giry_monad

group

haar_measure

integration

interval_integral

l1_space

lebesgue_measure

measurable_space

measure_space

outer_measure

probability_mass_function

set_integral

simple_func_dense

meta

coinductive_predicates

expr

expr_lens

rb_map

uchange

number_theory

basic

bernoulli

dioph

lucas_lehmer

pell

primorial

pythagorean_triples

quadratic_reciprocity

sum_four_squares

sum_two_squares

order

category

LinearOrder

NonemptyFinLinOrd

PartialOrder

Preorder

filter

at_top_bot

bases

basic

cofinite

countable_Inter

extr

filter_product

germ

indicator_function

interval

lift

partial

pointwise

ultrafilter

basic

boolean_algebra

bounded_lattice

bounds

complete_boolean_algebra

complete_lattice

conditionally_complete_lattice

copy

directed

fixed_points

galois_connection

ideal

iterate

lattice

lexicographic

liminf_limsup

ord_continuous

order_iso_nat

pilex

rel_classes

rel_iso

semiconj_Sup

zorn

representation_theory

maschke

ring_theory

ideal

basic

operations

over

polynomial

basic

homogeneous

rational_root

valuation

basic

adjoin

adjoin_root

algebra

algebra_operations

algebra_tower

algebraic

coprime

derivation

discrete_valuation_ring

eisenstein_criterion

euclidean_domain

fintype

fractional_ideal

free_comm_ring

free_ring

integral_closure

integral_domain

jacobson

jacobson_ideal

localization

maps

matrix_algebra

multiplicity

noetherian

polynomial_algebra

power_series

prime

principal_ideal_domain

subring

subsemiring

tensor_product

unique_factorization_domain

set_theory

game

domineering

impartial

nim

short

state

winner

cardinal

cardinal_ordinal

cofinality

game

lists

ordinal

ordinal_arithmetic

ordinal_notation

pgame

schroeder_bernstein

surreal

zfc

system

random

basic

tactic

converter

apply_congr

binders

interactive

old_conv

linarith

datatypes

elimination

frontend

lemmas

parsing

preprocessing

verification

lint

basic

default

frontend

misc

simp

type_classes

monotonicity

basic

interactive

lemmas

nth_rewrite

basic

congr

default

omega

int

dnf

form

main

preterm

nat

dnf

form

main

neg_elim

preterm

sub_elim

clause

coeffs

eq_elim

find_ees

find_scalars

lin_comb

main

misc

prove_unsats

term

rewrite_all

basic

abel

algebra

alias

apply

apply_fun

auto_cases

binder_matching

cache

cancel_denoms

chain

choose

clear

core

dec_trivial

delta_instance

derive_fintype

derive_inhabited

doc_commands

elide

equiv_rw

explode

ext

fin_cases

find

find_unused

finish

fix_reflect_string

generalize_proofs

generalizes

group

hint

interactive

interactive_expr

interval_cases

lift

local_cache

localized

mk_iff_of_inductive_prop

noncomm_ring

norm_cast

norm_num

obviously

pi_instances

protected

push_neg

rcases

reassoc_axiom

rename_var

replacer

restate_axiom

rewrite

ring

ring2

ring_exp

scc

show_term

simp_command

simp_result

simp_rw

simpa

simps

slice

solve_by_elim

split_ifs

squeeze

subtype_instance

suggest

tauto

tfae

tidy

transfer

transform_decl

transport

trunc_cases

unfold_cases

where

with_local_reducibility

wlog

zify

topology

algebra

affine

continuous_functions

group

group_completion

infinite_sum

module

monoid

multilinear

open_subgroup

ordered

polynomial

ring

uniform_group

uniform_ring

category

Top

adjunctions

basic

epi_mono

limits

open_nhds

opens

TopCommRing

UniformSpace

instances

complex

ennreal

nnreal

real

real_vector_space

metric_space

antilipschitz

baire

basic

cau_seq_filter

closeds

completion

contracting

emetric_space

gluing

gromov_hausdorff

gromov_hausdorff_realized

hausdorff_distance

isometry

lipschitz

pi_Lp

premetric_space

sheaves

presheaf

presheaf_of_functions

sheaf

sheaf_of_functions

stalks

uniform_space

absolute_value

abstract_completion

basic

cauchy

compact_separated

compare_reals

complete_separated

completion

pi

separation

uniform_convergence

uniform_embedding

bases

basic

bounded_continuous_function

compact_open

compacts

constructions

continuous_map

continuous_on

dense_embedding

extend_from_subset

homeomorph

list

local_extr

local_homeomorph

maps

opens

order

path_connected

separation

sequences

stone_cech

subset_properties

tactic

topological_fiber_bundle